Wave functions are mathematical descriptions of quantum particles, crucial for understanding their behavior. They allow us to calculate probability distributions, revealing where particles are likely to be found in space.

The Born interpretation connects wave functions to measurable probabilities. This section covers key concepts like probability density, radial distribution, normalization, orthogonality, and parity, essential for grasping quantum mechanics' foundations.

Probability and Interpretation

Probability Density and Born Interpretation

• Probability density $|\Psi(x)|^2$ represents the probability of finding a particle at a specific location $x$
• Born interpretation states that the probability of finding a particle in a given region is equal to the integral of the probability density over that region
  • Probability of finding a particle between $x_1$ and $x_2$ is given by $\int_{x_1}^{x_2} |\Psi(x)|^2 dx$
• The probability density is always non-negative and real-valued
• In three dimensions, the probability density is given by $|\Psi(x, y, z)|^2$

Radial Distribution Function and Normalization

• Radial distribution function $P(r)$ describes the probability of finding an electron at a distance $r$ from the nucleus in an atom
  • Calculated using the wave function as $P(r) = r^2 |\Psi(r)|^2$
  • Useful for understanding the spatial distribution of electrons in an atom (s, p, d, f orbitals)
• Normalization ensures that the total probability of finding a particle over all space is equal to 1
  • Mathematically, $\int_{-\infty}^{\infty} |\Psi(x)|^2 dx = 1$ for a normalized wave function
  • Normalization is necessary for the Born interpretation to be valid
  • Wave functions can be multiplied by a constant to achieve normalization

Wave Function Properties

Orthogonality and Nodes

• Orthogonality is a property of two or more wave functions that indicates they are independent and do not overlap
  • Mathematically, two wave functions $\Psi_1$ and $\Psi_2$ are orthogonal if $\int \Psi_1^ \Psi_2 dx = 0$
  • Orthogonal wave functions represent different quantum states of a system (energy levels in an atom)
• Nodes are points or regions where the wave function equals zero
  • The number of nodes in a wave function is related to the energy of the state (higher energy states have more nodes)
  • Nodes occur between regions of positive and negative values of the wave function
  • Example: The 2s orbital in a hydrogen atom has one node, while the 3s orbital has two nodes

Parity and Symmetry

• Parity refers to the symmetry of a wave function under spatial inversion (changing the sign of all coordinates)
  • Even parity: $\Psi(-x) = \Psi(x)$, the wave function is symmetric (s, d orbitals)
  • Odd parity: $\Psi(-x) = -\Psi(x)$, the wave function is antisymmetric (p, f orbitals)
• Parity is conserved in systems with symmetric potentials, such as the harmonic oscillator or the hydrogen atom
• The parity of a wave function has implications for the allowed transitions between quantum states (selection rules)